Fractional Decomposition Calculator

Formula

The formula to calculate the fractional decomposition of a rational function is:

$\frac{A}{B} = \frac{A1}{B1} + \frac{A2}{B2} + \cdots + \frac{An}{Bn}$

Where:

$\frac{A}{B}$ is the original rational function
$A1, A2, \ldots, An$ are the numerators of the decomposed fractions
$B1, B2, \ldots, Bn$ are the denominators of the decomposed fractions

What is Fractional Decomposition?

Fractional decomposition is a mathematical process used to break down complex fractions or rational expressions into simpler parts, often for the purpose of integration or simplification. It involves expressing the fraction as a sum of simpler fractions with linear or quadratic denominators. This method is particularly useful in calculus and algebra for solving equations and integrating functions.

Example Calculation

Let's assume the following rational function:

$\frac{2x + 3}{(x - 1)(x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2}$

To find $A$ and $B$ , we multiply both sides by $(x - 1)(x + 2)$ :

$2x + 3 = A(x + 2) + B(x - 1)$

Next, we solve for $A$ and $B$ by setting up equations for the coefficients:

For $x$ : $2 = A + B$

For the constant term: $3 = 2A - B$

Solving these equations gives us $A = 1$ and $B = 1$ .

Thus, the fractional decomposition is:

$\frac{2x + 3}{(x - 1)(x + 2)} = \frac{1}{x - 1} + \frac{1}{x + 2}$